- #1
Gunthi
- 65
- 1
Homework Statement
[tex]\mathcal{L}\left[\frac{\sin(t)}{t}\right]=?[/tex]
Homework Equations
[tex]\mathcal{L}\left[f(t)\right]=\int_0^\infty e^{-zt}f(t)dt[/tex]
The Attempt at a Solution
[tex]\int_0^\infty e^{-zt}\frac{\sin(t)}{t}dt=\int_0^\infty \frac{e^{-zt}}{t}\left(\frac{e^{it}-e^{-it}}{2i}\right )dt[/tex]
[tex]\int_0^\infty f(t)dt=\lim_{T\rightarrow\infty}\int_0^T \frac{1}{t}\left(\frac{e^{t(i-z)}-e^{-t(i+z)}}{2i}\right )[/tex]
And I end up having to calculate things like
[tex]\int_0^T\frac{e^{\alpha t}}{t}dt[/tex] which doesn't seem to be the best way to do this.
Does anyone know how to do this properly?
Any help would be appreciated, thanks in advance!